In order to understand the mass hierarchy problem in the neutrino sector, there are many models in the literature to naturally generate small Majorana neutrino masses such as the traditional Seesaw [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, [8]}, [9]}, [10]}, [11]}, [12]}, [13]} and radiative mass generation mechanisms [14]}, [15]}, [16]}, [17]}, [18]}, [19]}, [20]}. Most of them can be summarized as a specific realization of the conventional dimension-5 Weinberg operator. However, the generation of Majorana neutrino masses only requires the lepton number violation (LNV) by two units, and there exist many other equally legitimate LNV effective operators [21]}, [22]}, [23]}, [24]}, [25]}, [26]}, [27]}, [28]}, [29]}, which are composed of the SM fields but with higher scaling dimensions. From the effective field theory perspective, it is generically believed that these high-dimensional effective operators are subdominated by the Weinberg operator due to the suppression from the corresponding high powers of the large cutoff. In order for these operators to show up as the leading contributions, one usually needs to impose an additional symmetry on the model to break the usual scaling arguments. Furthermore, if this symmetry is kept unbroken, then the lightest symmetry-protected neutral particle would provide a perfect DM candidate. In this way, the symmetry connects Majorana neutrino masses and DM physics by the high-dimensional effective operators. Such a connection
has been already exemplified by some recent three-loop neutrino mass models [17]}, [31]}, [32]}, [33]}, which realize the dimension-9 effective operator \({\cal O}_9 = \overline{l^c}_{R}l_{R} [(D_\mu \Phi ) \Phi ]^2\)  with the DM embedded in the loop.
